consider the function f given below. f(x) = (x^2 - 9)/(x - 3) a) for what x - values(s) is this function not…

consider the function f given below. f(x) = (x^2 - 9)/(x - 3) a) for what x - values(s) is this function not differentiable? b) find f’(9). a) f(x) is not differentiable at x = . (use a comma to separate answers as needed.) b) f’(9) = .

consider the function f given below. f(x) = (x^2 - 9)/(x - 3) a) for what x - values(s) is this function not differentiable? b) find f’(9). a) f(x) is not differentiable at x = . (use a comma to separate answers as needed.) b) f’(9) = .

Answer

Explanation:

Step1: Simplify the function

First, factor the numerator $x^{2}-9=(x + 3)(x - 3)$. Then $f(x)=\frac{(x + 3)(x - 3)}{x - 3}=x + 3,x\neq3$. The function has a removable - discontinuity at $x = 3$. A function is not differentiable at points of discontinuity.

Step2: Find the derivative of the simplified function

The derivative of $y=x + 3$ using the power rule $\frac{d}{dx}(x^{n})=nx^{n - 1}$ (for $n = 1$) is $f'(x)=\frac{d}{dx}(x+3)=1$.

Step3: Evaluate $f'(9)$

Since $f'(x)=1$ for all $x\neq3$, then $f'(9)=1$.

Answer:

a) 3 b) 1